Piezoelectric actuators have been widely used for precise multidimensional positioning from subnanometer to micrometer scale due to their high stiffness, compact size, and high dynamic responses. The piezoelectric material is subject to dielectric hysteresis resulting in a nonlinear relationship between the applied voltage and the output mechanical displacement. Some approaches have been used to overcome the hysteresis, including feed forward control using inverse model of hysteresis, such as the Preisach model, domain wall theory and homogenized energy model. However, these models have shortcomings of being sophisticated, lack of robustness, plant parameter uncertainty, and being computationally cumbersome. Other approaches, including feedback of displacement control and feedback of induced charge, are limited by high cost or poor sensor performance.
The basic configuration of the traditional voltage control circuit is shown in FIG 1a. The piezoelectric actuator is modeled as a capacitor Cact. The voltage controller is an inverting amplifier, which magnifiers the input voltage to drive the piezoelectric actuator.
The basic configuration of the traditional charge amplifier is shown in FIG 1b. In FIG 1b, the charge controller is an inverting amplifier, which magnifies and converts the input voltage to proportional charges on the actuator at high frequencies.
Assuming that both the actuator and the input capacitor are initially discharged and the op-amp is ideal, in FIG 1a, the driving voltage exerted on the piezoelectric actuator is presented as
                              V          o                =                              -                                          R                f                                            R                1                                              ⁢                                    V              in                        .                                              (        1        )            In FIG 1b, supposing the input capacitor C1 is ideal, the driving voltage can be expressed as
                                          V            o                    ⁡                      (                          j              ⁢                                                          ⁢              ω                        )                          =                              -                                          R                f                                            R                1                                              ⁢                                                                      R                  1                                ⁢                                  C                  1                                ⁢                j                ⁢                                                                  ⁢                ω                            +              1                                                      R                f                            ⁢                              C                act                            ⁢              j              ⁢                                                          ⁢              ω                                ⁢                                                    V                in                            ⁡                              (                                  j                  ⁢                                                                          ⁢                  ω                                )                                      .                                              (        2        )            Assuming thatR1C1=RfCact,  (3)then the voltage gain is a constant and independent of frequency. The amplifier can be viewed as a concatenation of the voltage and charge amplifiers. For input frequencies ω1/R1C1, the capacitances are negligible, and as such, the resistances determine the transfer function as
                                          V            o                    =                                    -                                                R                  f                                                  R                  1                                                      ⁢                          V              in                                      ,                            (        4        )            which is the same as expression (1). In this case, the voltage Vo on the actuator is proportional to the input voltage Vin, and the circuit operates as a voltage controller. Therefore, at low frequencies, the hysteresis of the actuator causes nonlinear operation. On the other hand, for input frequencies ω1/R1C1, the resistances are negligible. Thus, the capacitances determine the actuator voltage according to
                                          V            0                    =                                                    -                                                      C                    1                                                        C                    act                                                              ⁢                              V                in                                      =                                          Q                act                                            C                act                                                    ,                            (        5        )            and as such, the circuit acts as a charge controller. Therefore, at high frequencies, the actuator has good linear characteristics. It should be noted that Vo is not proportional to Vin, given that the equivalent actuator capacitance changes nonlinearly with the voltage applied on it. If R1 and Rf are infinitely large, the charges on the capacitor C1 can be transferred totally to the piezoelectric actuator at relatively low frequencies. In this case, the quantity of the charges delivered to the actuator is linear with the input voltage Vin, and the charge in the Cact has no influence on actuator performance.
The above described expressions are based on the presumption that the input capacitor is ideal. In this charge controller, a capacitor with a much larger capacitance than the actuator is required to ensure proper operation of the circuit. However, a large capacitor usually has large dielectric absorption, indicating that the input capacitor C1 is not a pure capacitor but one with nonignorable resistance component, as shown in the inset of FIG 1b. 
Therefore, for existing charge driver, the two factors that may influence its performances are the resistor network and the dielectric absorption of the input capacitor. It is not a charge driving system and has inevitable influence on the performance of the actuators. The existence of the resistor network worsens displacement linearity at low frequencies, while the dielectric absorption causes nonlinear displacement response over all the frequency ranges.